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Hi I have the following matrix

A=[a_11 a_12 a_13 1; a_21 a_22 a_23 1; . . . a_n1 a_n2 a_n3 1]

I have seen that when some of a_ij are big for instance in the order of 200 , then condition number is also big.

I would like to know is it possible to show it theoretically.

Is it possible to find a lower bound for this problem?



share|cite|improve this question discusses lower bounds for the condition number – stankewicz Sep 27 '10 at 19:37
up vote 1 down vote accepted

One possible way to proceed would be to get the singular value decomposition of your matrix and then look at the ratio of the largest singular value to the smallest singular value (a.k.a. the 2-norm condition number); largeness of this condition number implies largeness of the condition number with respect to the other norms of interest, and you can probably just manipulate inequalities at that point.

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Having large entries is not related at all to the condition number. For example, the matrix $10,000I$ (ten thousand times the identity) has condition number 1. One can easily generate matrices with arbitrarily large and small entries (even at the same time) and condition number 1.

In order to prove a bound, you need to exploit some more structural properties of the matrix you are working with.

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